Let $X_t:=\sqrt t B_1$ and $Y_t:=B_{2t}-B_t$ where $(B_t)_{t\geq0}$ is a Brownian motion.
Why $(X_t)_{t\geq0}$ and $(Y_t)_{t\geq0}$ are not a Brownian motion ?
2026-03-28 15:36:43.1774712203
Why $X_t:=\sqrt t B_1$ and $Y_t:=B_{2t}-B_t$ are not a Brownian motion?
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$X_1$ and $X_2$ (or any other pair) is perfectly corelated and they shouldn't be. Similarly $Y_1$ and $Y_2$ are independent (as they are the increments of a BM on disjoint intervals) and they shouldn't be (so they have corelation $0$). $B_1$ and $B_2$ should have correlation $\frac{1}{\sqrt2}$.